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Every Point on the Bisector of an Angle Is Equidistant from the Sides of the Angle.

The combined ...

Question

The combined equation of the pair of lines through the point $(1, 0)$ and parallel to the lines represented by $2 x^{2} - x y - y^{2} = 0$ is

2 x^{2} - x y - 2 y^{2} + 4 x - y = 6

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2 x^{2} - x y - y^{2} - 4 x - y + 2 = 0

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2 x^{2} - x y - y^{2} - 4 x + y + 2 = 0

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Solution

The correct option is B $2 x^{2} - x y - y^{2} - 4 x + y + 2 = 0$
$2 x^{2} - x y - y^{2} = 0$
By factorization,
$(2 x + y) (x - y) = 0$
Since, we have given that combined equation of the pair is parallel to $2 x^{2} - x y - y^{2} = 0$
So combined equation will be,
$(2 x + y + k_{1}) (x - y + k_{2}) = 0$
This equation satisfies $(1, 0)$
$(2 \times 1 + 0 + k_{1}) (1 - 0 + k_{2}) = 0$
$k_{2} = - 1, k_{1} = - 2$
Put $k_{1}, k_{2}$ in main equation and get combined equation as
$2 x^{2} - x y - y^{2} - 4 x + y + 2 = 0$

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